Project Title: A computerized investigation of numerical ranges

Advisor: Valentin Matache

Description: The domain of definition of a function is in general easy to specify. Its range is generally upper estimated and rarely exactly described. It is of interest for mathematicians, physicists, and engineers to understand the image of the unit sphere (of a space or another) under the quadratic form associated to a matrix, (finite or not), called the numerical range of that matrix. The first part of this project requires that the student becomes familiar with the basic properties of numerical ranges e.g., as summarized by this advisor in:

http://avalon.unomaha.edu/~wwwmath/MAM/2002/Poster02/Sample_Poster.pdf

The second task to be performed by the student is becoming familiar with MATLAB, (only the basic skills necessary to run routines provided by the advisor are needed). Next, the student will run the aforementioned routines for numerous classes of matrices as instructed by the advisor and summarize his/her observations. The final task is to create a final report to be submitted to the Math Department and presented in a 20 minutes talk during the MAM Symposium in April 2008.

OTHER REQUIREMENTS: The students interested in the project above are expected to have had an Introductory Course in Analysis, respectively Linear Algebra, and understand well computations with complex numbers.